Theorem ialgrlem1st 12613

Description: Lemma for ialgr0 12615. Expressing algrflemg 6394 in a form suitable for theorems such as seq3-1 10723 or seqf 10725. (Contributed by Jim Kingdon, 22-Jul-2021.)

Hypothesis

Ref	Expression
ialgrlem1st.f	⊢ (𝜑 → 𝐹:𝑆⟶𝑆)

Assertion

Ref	Expression
ialgrlem1st	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(𝐹 ∘ 1st )𝑦) ∈ 𝑆)

Proof of Theorem ialgrlem1st

Step	Hyp	Ref	Expression
1		algrflemg 6394	. . 3 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥(𝐹 ∘ 1st )𝑦) = (𝐹‘𝑥))
2	1	adantl 277	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(𝐹 ∘ 1st )𝑦) = (𝐹‘𝑥))
3		ialgrlem1st.f	. . . 4 ⊢ (𝜑 → 𝐹:𝑆⟶𝑆)
4	3	adantr 276	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝐹:𝑆⟶𝑆)
5		simprl 531	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑥 ∈ 𝑆)
6	4, 5	ffvelcdmd 5783	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝐹‘𝑥) ∈ 𝑆)
7	2, 6	eqeltrd 2308	1 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(𝐹 ∘ 1st )𝑦) ∈ 𝑆)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1397   ∈ wcel 2202   ∘ ccom 4729  ⟶wf 5322  ‘cfv 5326  (class class class)co 6017  1^st c1st 6300

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fo 5332  df-fv 5334  df-ov 6020  df-1st 6302

This theorem is referenced by:  ialgr0  12615  algrp1  12617